Markov degree of the three-state toric homogeneous Markov chain model

TL;DR

This paper analyzes the geometric and algebraic structure of the three-state toric homogeneous Markov chain model, revealing its polytope facets and bounding the degree of its toric ideal generators.

Contribution

It provides a complete description of the model polytope's facets and proves the normality of the associated toric variety, leading to bounds on the ideal's generator degrees.

Findings

The model polytope has 24 facets for all T ≥ 5.

The toric ideal is generated by binomials of degree at most 6.

The associated toric variety is normal.

Abstract

We consider the three-state toric homogeneous Markov chain model (THMC) without loops and initial parameters. At time $T$, the size of the design matrix is $6 \times 3\cdot 2^{T-1}$ and the convex hull of its columns is the model polytope. We study the behavior of this polytope for $T\geq 3$ and we show that it is defined by 24 facets for all $T\ge 5$. Moreover, we give a complete description of these facets. From this, we deduce that the toric ideal associated with the design matrix is generated by binomials of degree at most 6. Our proof is based on a result due to Sturmfels, who gave a bound on the degree of the generators of a toric ideal, provided the normality of the corresponding toric variety. In our setting, we established the normality of the toric variety associated to the THMC model by studying the geometric properties of the model polytope.